The space \(\mathfrak {s}\mathfrak {vect}^*(M)\)  of regular functionals on \(\mathfrak {s}\mathfrak {vect}(M)\)  is a dense subset in the space of all continues linear functionals on \(\mathfrak {s}\mathfrak {vect}(M).\)  It turns out that the space of regular functionals \(\mathfrak {s}\mathfrak {vect}^*(M)\)  can be identified with the space of cosets
\(\Omega ^1(M)/d\Omega ^0(M),\)  since exact 1-forms give zero functionals on divergence-free vector fields. Moreover, the natural action of the group \(\mathrm {SDiff}(M)\)  on the space of cosets \(\Omega ^1(M)/d\Omega ^0(M)\)  by means of pull-backs coincides with the coadjoint action of the group of symplectomorphisms \(\mathrm {SDiff}(M)\) . The proof of this fact can be found in [1]} (see Section I.8). More information about infinite-dimensional Lie groups can be found in [2]}.
